| L(s) = 1 | + 0.452·3-s + 1.91·5-s + 1.96·7-s − 2.79·9-s + 0.787·11-s + 5.79·13-s + 0.864·15-s − 3.13·17-s + 4.63·19-s + 0.887·21-s + 2.51·23-s − 1.35·25-s − 2.62·27-s + 3.62·29-s − 6.20·31-s + 0.356·33-s + 3.74·35-s + 9.71·37-s + 2.62·39-s − 0.892·41-s − 5.80·43-s − 5.34·45-s + 3.65·47-s − 3.15·49-s − 1.41·51-s + 12.6·53-s + 1.50·55-s + ⋯ |
| L(s) = 1 | + 0.261·3-s + 0.854·5-s + 0.741·7-s − 0.931·9-s + 0.237·11-s + 1.60·13-s + 0.223·15-s − 0.759·17-s + 1.06·19-s + 0.193·21-s + 0.523·23-s − 0.270·25-s − 0.504·27-s + 0.673·29-s − 1.11·31-s + 0.0620·33-s + 0.633·35-s + 1.59·37-s + 0.419·39-s − 0.139·41-s − 0.884·43-s − 0.796·45-s + 0.533·47-s − 0.450·49-s − 0.198·51-s + 1.73·53-s + 0.202·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.547487905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.547487905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 0.452T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 0.787T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 41 | \( 1 + 0.892T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 6.08T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 - 7.14T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 4.55T + 83T^{2} \) |
| 89 | \( 1 - 1.55T + 89T^{2} \) |
| 97 | \( 1 - 5.55T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.955185322554819080436254744126, −8.589616125670388730183832210399, −7.72922694852428847458512444674, −6.69573627829248048118857117084, −5.85943706106730206860664857062, −5.35628403896775333928560353168, −4.19761342455787752396370874983, −3.20976778604699857323626861705, −2.17962911494779390665286791396, −1.14806160919623539000309059604,
1.14806160919623539000309059604, 2.17962911494779390665286791396, 3.20976778604699857323626861705, 4.19761342455787752396370874983, 5.35628403896775333928560353168, 5.85943706106730206860664857062, 6.69573627829248048118857117084, 7.72922694852428847458512444674, 8.589616125670388730183832210399, 8.955185322554819080436254744126
